According to the GIRM:
- ...[Extraordinary Ministers of Holy Communion] should not approach the altar before the priest has received Communion, and they are always to receive from the hands of the priest celebrant the vessel containing either species of the Most Holy Eucharist for distribution to the faithful.
So, no, they should not be receiving communion at the same time as the priest since they should not be in the sanctuary at that time.
As for purification the GIRM states:
- The sacred vessels are purified by the priest, the deacon, or an instituted acolyte after Communion or after Mass, insofar as possible at the credence table.
So once again, the answer is no, EMHC should not purify the sacred vessels.